Residual-based a posteriori error analysis of an ultra-weak discontinuous Galerkin method for nonlinear second-order initial-value problems

In this paper, we propose and analyze implicit residual-based a posteriori error estimates for the ultra-weak discontinuous Galerkin (UWDG) method for nonlinear second-order initial-value problems for ordinary diﬀerential equations of the form u ′′ = f ( x, u ). We prove that the UWDG error on each element can be split into two parts. The significant part is proportional to the ( p + 1)-degree polynomial (1 − ξ ) 2 P 2 , 0 p − 1 ( ξ ) , ξ ∈ [ − 1 , 1], where P 2 , 0 p − 1 ( ξ ) is the ( p − 1)-degree Jacobi polynomial, when piecewise polynomials of degree at most p ≥ 2 are used. The second part of the error converges with order p + 2 in the L 2 -norm. These results allow us to construct a posteriori UWDG error estimates. The proposed residual-based a posteriori error estimators of this paper are reliable, eﬃcient, and are obtained by solving a local problem with no side conditions on each element. Furthermore, we prove that, for smooth solutions, these a posteriori error estimates converge to the exact errors in the L 2 -norm under mesh reﬁnement. The order of convergence is proved to be p + 2. Finally, we prove that the global eﬀectivity index converges to unity at O ( h ) rate. Several numerical experiments are provided to validate the theoretical results.


Introduction
This paper proposes and analyzes reliable and efficient residual-based a posteriori error estimators for the ultra-weak discontinuous Galerkin (UWDG) method for the following nonlinear second-order initial-value problem (IVP) where f : [a, b] × R → R is a given smooth function.The specific assumptions needed in our error analysis will be given in Section 2.6.
This work is a continuation of our paper [9] in which the convergence and superconvergence properties of an UWDG method for nonlinear second-order initial-value problems (IVPs) for ordinary differential equations (ODEs) of the form (1.1) is studied.In [9], we used a suitable choice of the numerical fluxes to prove optimal L 2 error estimates for the UWDG solution.
To be more specific, we proved that the UWDG solution u h is (p + 1)-th order convergent in the L 2 -norm, when piecewise polynomials of degree at most p ≥ 2 are used.We further established a supercloseness result in the L 2 -norm towards a special projection of the exact solution.We proved that the UWDG solution is superconvergent with order p + 2 for p = 2 and p + 3 for p ≥ 3 towards the special projection of the exact solution P − h u.Finally, we proved that the p-degree UWDG solution and its derivative are O(h 2p ) superconvergent at the end of each step.In this work we use the results in [9] to construct efficient and reliable a posteriori error estimates for the UWDG method.We believe that this is the first error analysis carried out for the UWDG method for the IVP (1.1).
A posteriori error estimates play an essential role in assessing the reliability of numerical solutions and in developing efficient adaptive algorithms.A posteriori error estimates are traditionally used to guide adaptive enrichment by h-and p-refinement and to provide a measure of solution reliability.Typically, a posteriori error estimators employ the known numerical solution to derive estimates of the actual solution errors.
In recent years, the study of superconvergence and a posteriori error estimates of DG methods has been an active research field in numerical analysis, see the monographs by Verfürth [16], Wahlbin [17], and Babuška and Strouboulis [5].A knowledge of superconvergence properties can be used to (i) construct simple and asymptotically exact a posteriori estimates of discretization errors and (ii ) help detect discontinuities to find elements needing limiting, stabilization and/or refinement.A posteriori error estimates play an essential role in assessing the reliability of numerical solutions and in developing efficient adaptive algorithms.Typically, a posteriori error estimators employ the known numerical solution to derive estimates of the actual solution errors.They are also used to steer adaptive schemes where either the mesh is locally refined (h-refinement) or the polynomial degree is raised (p-refinement).For an introduction to the subject of a posteriori error estimation see the monograph of Ainsworth and Oden [4].Superconvergence properties for finite element and DG methods for ordinary differential equations have been studied in [2,3,6,7,8,11,12,18,15].The first superconvergence result for standard DG solutions of hyperbolic PDEs appeared in Adjerid et al. [2].The authors presented numerical results that show that standard DG solutions of one-dimensional linear and nonlinear hyperbolic problems using p-degree polynomial approximations exhibit an O(h p+2 ) superconvergence rate at the roots of (p + 1)-degree Radau polynomial.They further established a strong O(h 2p+1 ) superconvergence at the downwind end of every element.
In [3], Adjerid and Temimi proposed and analyzed a new UWDG finite element method to solve linear IVPs for ODEs.They performed a local error analysis to show that the UWDG solution exhibits an optimal convergence rate in the L 2 -norm.The order of convergence is proved to be of order p + 1, when piecewise polynomials of degree at most p are used.They further showed that the p-degree UWDG solution of m-th order ODEs and its first m − 1 derivatives are superconvergent with order 2p + 2 − m at the end of each step.Moreover, they established that the p-degree discontinuous solution is superconvergent with order p + 2 at the roots of (p + 1 − m)-degree Jacobi polynomial on each step.Finally, as an application of the superconvergence results, they constructed asymptotically exact a posteriori error estimates.Their computational results indicate that the error estimates converge to the true error under both h-refinement.However, even for linear IVPs, there is no theoretical justification of these results so far.
In this paper, we present and analyze an implicit a posteriori UWDG error estimate for the model IVP (1.1).We use the results of the first part of this work [9] to prove that the significant part of the spatial discretization error for the UWDG solution is proportional to the (p + 1)-degree Jacobi polynomial, when piecewise polynomials of degree at most p are used.We use this result to construct a residual-based a posteriori error estimate for the spatial error.The leading term of the discretization error is estimated by solving a local problem with no boundary conditions on each element.We further prove that the proposed UWDG error estimate converges to the true error at O(h p+2 ) rate.Finally, we prove that the global effectivity index in the L 2 -norm converges to unity at O(h) rate.Our proofs are valid for any regular meshes and using piecewise polynomials of degree p ≥ 2. This paper is organized as follows.In Section 2, we recall the the UWDG method for the second-order IVP (1.1).We also introduce some basic notations and preliminaries which will be used later.In Section 3, we present new superconvergence results.We present our a posteriori error estimation procedure and prove that these error estimates converge to the true errors under mesh refinement in L 2 -norm with optimal convergence rate in Section 4. Numerical examples are provided to show the accuracy and capability of the scheme in Section 5. Some concluding remarks and future work are given in Section 6.

The UWDG method and Preliminaries
This section is devoted to the definition of the UWDG method.We also provide some notation, projections, the a priori error estimates, and superconvergence results provided in [9].

The UWDG scheme
To define the UWDG method, we divide [a, b] into N intervals h i be the length of I i and the length of the largest interval, respectively.We assume that the mesh is regular, namely the ratio between the maximum and the minimum mesh sizes stays bounded during mesh refinements.
Multiplying the ODE in (1.1) by a test function v, integrating over I i , and using integration by parts twice, we get the UWDG weak formulation To construct the UWDG scheme, we introduce the following discontinuous finite element approximation space V p h = {v : v| I i ∈ P p (I i ), i = 1, 2, . . ., N }, where P p (I i ) denotes the space of polynomials of degree at most p on the interval I i .
We denote by v(x + i ) and v(x − i ) the values of v at x = x i from the right cell I i+1 and from the left cell I i , respectively, i.e.,

v(x
Next, we use the weak formulation (2.1) to define the UWDG scheme.Find u h ∈ V p h such that for all test function v ∈ V p h , the following formulation holds: for all i = 1, 2, . . ., N .Here u h and u ′ h are the numerical fluxes.They are the boundary terms that emerge from integration by parts.These numerical fluxes should be designed based on different guiding principles for different equations to guarantee stability and convergence.
To completely define the UWDG scheme we must choose the numerical fluxes u h and u ′ h .We emphasize that the choice of the numerical fluxes u h and u ′ h is crucial for the accuracy of the UWDG method.To define these numerical fluxes, let us first introduce some notation.We define the mean values {{•}} and jumps • of a function v ∈ V p h at the point x i as follows: Now, we are ready to define the numerical fluxes.In this paper, we take the numerical fluxes which depend on the boundary conditions (2.2b)After we defined the numerical fluxes, the discrete problem (2.2) becomes an algebraic system of nonlinear equations for the unknown coefficients in u h .This system can be solved numerically by one of the classical methods for solving nonlinear systems of equations such as Newton's method.

Norms
We will use the following notation throughout the paper: for s ≥ 0, is the H s (I i )-norm for a real-valued function u ∈ H s (I i ), is the L 2 -norm of u on the whole domain Ω, for s ≥ 1, is the H s -norm of u on the whole domain Ω, • |u| s,I i = ∥D s u∥ 0,I i is the semi-norm on I i , is the semi-norms on the whole domain Ω.

Projections
For p ≥ 1, we define the special projection P − h into V p h such that, for any u, the projection P − h u satisfies [9]: for all i = 0, 1, . . ., N Similarly, we need the special projection P + h into V p h which satisfies For the above one-dimensional projection, the following a priori error estimates hold [9] u where Γ h denotes the set of boundary points of all elements I i , and C is a positive constant dependent on p but not on h.

Inverse properties
Here, we recall the classical inverse properties of the finite element space V p h [10].We summarize them in the following lemma.Lemma 2.1.Suppose that h then there exists a positive constant C independent of the mesh size h such that

Jacobi polynomials
In our analysis we will use the classical Jacobi polynomial [1] defined by the Rodgrigues formula for ξ ∈ [−1, 1].We note that when α = β = 0, the Jacobi polynomial reduces to the kthdegree Legendre polynomial, which will be denoted by Pk (ξ) = P 0,0 k (ξ) on [−1, 1].We further note Jacobi polynomials satisfy the orthogonality relation where > 0 and δ kl is the Kronecker symbol equal to 1 if k = l and 0, otherwise.Here Γ(x) is the classic Gamma function.
The following properties will be needed in our error analysis: We also need the coefficient of the term ξ k in P α,β k (ξ) which is given by . Mapping the physical element I i into the reference element [−1, 1] by the standard affine mapping we get the k-degree shifted Jacobi polynomial P α,β k,i (x) and the k-degree shifted Legendre polynomial P k,i (x) on Throughout this paper the roots of the (p+1)-degree polynomial Tp+1 1] are denoted by ξ i , i = 0, 1, . . ., p and the roots of the (p + 1)-degree polynomial , j = 0, 1, . . ., p. (2.10) We note that the (p + 1)-degree polynomial T p+1,i (x) on I i can be written as The following properties of T p+1,i which will be needed in our a posteriori error analysis where T ′′ p+1 (ξ) Tp+1 (ξ) dξ are constants independent of the mesh size h.

A Priori error analysis
In this subsection, we recall the error estimates of our UWDG scheme for the model problem (1.1).From now on, we use C (without or with subscripts or superscripts) to denote a generic constant that is independent of the mesh size h.However, it can take different values in different places.All constants may depend on the smooth exact solution of the model problem (1.1).
Let e u be the error between the exact and UWDG solution defined by As the general treatment of the finite element methods, we split the actual error into two terms as where the projection error is defined by and ξ u ∈ V p h is the error between the UWDG solution and the projection of the exact solution i.e., ξ In our error analysis, we always assume that the function f (x, u) : D → R appearing in (1.1) is sufficiently differentiable function with respect to its arguments.To be more precise, we impose the following assumptions on f (x, u): Assumption A2.There exists a positive constant L such that ∂f (x, u) ∂u ≤ L, for all (x, u) ∈ D. (2.14) Using the Mean-Value Theorem, it follows that f satisfies the following uniform Lipschitz condition on D in the variable u with uniform Lipschitz constant (2.15) Now, we are ready to state our a priori error estimate for e u in the L 2 -norm.
Theorem 2.1.Suppose that u ∈ H p+1 (Ω) solves (1.1).Let p ≥ 2 and u h be the UWDG solution defined in (2.2).If the function f satisfies Assumption A1 and Assumption A2.Then, for sufficiently small h, there exists a positive constant C independent of the mesh size h such that (2.17) Next, we summarize our superconvergence towards P − h u and at the mesh points from [9] in the following theorem.
Theorem 2.2.Suppose that the assumptions of Theorem 2.1 are satisfied.We further assume that g(x) = ∂f (x,u(x)) ∂u ∈ C p (Ω) is a sufficiently smooth function.Then there exists a positive constant C independent of h such that Proof.See [9].More precisely, the proof of (2.18) can be found in its Theorem 4.1 and the proofs of (2.19)-(2.22)can be found in its Theorem 4.2.

Superconvergence towards Jacobi interpolating polynomial
In this section, we use the results of Theorems 2.1 and 2.2 to show that the true error e u can be divided into two parts: a significant part and a less significant part.The significant part is proportional to a (p + 1)-degree polynomial (1 − ξ) 2 P 2,0 p−1 (ξ) where P 2,0 p−1 (ξ) is the (p − 1)-degree Jacobi polynomial.The less significant part converges at O(h p+2 ) rate in the L 2 -norm.
First, we define two interpolation operators π and π.The operator π is defined as follows: For any function u = u(x), πu| I i ∈ P p (I i ) and interpolates u at the roots x i,j , j = 0, 1, . . ., p, of the (p + 1)-degree polynomial T p+1 (x) on I i , i.e., at the nodes . The operator π is such that πu| I i ∈ P p+1 (I i ) and is defined as follows: πu| I i interpolates u at x i,j , j = 0, 1, . . ., p, and at an additional point xi in I i with xi ̸ = x i,j , i = 0, 1, . . ., p.For simplicity we choose xi = x i−1 .
Remark 3.1.The operator π is needed for technical reasons in the proof of the error estimates.We would like to mention that the interpolating polynomial πu depends on the additional point xi .We note that we can easily verify the following πu = πu + c p+1 T p+1,i (x).
(3.1) Using (3.1) and the fact that πu In the next lemma, we prove some properties of T p+1,i (x) which will be needed in our a posteriori error analysis.In particular, we show that the interpolation error can be divided into significant and less significant parts.
Lemma 3.1.Let P − h u be the projection defined in (2.3) and πu be the interpolating polynomial that interpolated u at x i,j , j = 0, 1, . . ., p,.Then Moreover, if u ∈ H p+2 (I i ) then the interpolation error u − πu can be split as: where a i is the coefficient of T p+1,i (x) in the (p + 1)-degree polynomial πu and ) Finally, we have the following superconvergence result where v 1 ∈ P p (I i ) and d i is a constant.Applying the operators π and P − h and using the fact that πv Using the standard interpolation error formula and (2.11), there exists y ∈ I i such that the interpolation error T p+1,i − π(T p+1,i ) is On the other hand, since P − h (T p+1,i ) ∈ P p (I i ), it can be written as Multiplying (3.8) by P k,i (x), k = 0, . . ., p − 2, integrating over I i , using the orthogonality property of the projection P − h , and applying the relation (2.8), we obtain, for k = 0, 1, . . ., p−2, 0 = Consequently b k = 0 for k = 0, 1, . . ., p − 2 so that By the property of the projection (3.11b) since P k,i (x i ) = 1 and P ′ k,i (x i ) = k(k+1) h i .Solving the system in (3.11), we get b p−1 = b p = 0. Therefore, Combining (3.6) and (3.12), we obtain From (3.7) and (3.13), we establish that Next, we prove (3.3).Adding and subtracting V = πu = p k=0 split the interpolation error as We note that, by (3.1), πu = π(πu) = πV .Thus, by (3.12), we get Multiplying πu = p k=0 a k P k,i +a i T p+1,i by P p+1,i , integrating over I i , and using the orthogonality relation (2.8), we obtain which gives Thus, we completed the proof of (3.3).
Next, we will prove (3.4).By the standard interpolation error estimates we have Finally, we find a bound of ∥V ∥ p+1,I i by adding and subtracting u and applying the triangle inequality as which complete the proofs of (3.4).
In order to prove (3.5) we note that πu ∈ P p+1 (I i ), thus by (3.2) and (3.1), we have and by the standard interpolation error we have Applying P − h to u = u − πu + πu and using (3.15), we obtain which, in turn, yields Now, we show that P − h v 0,I i ≤ C 2 ∥v∥ 0,I i by writing (3.18) Taking the L 2 norm of (3.17) and applying the estimate (3.18) with v = u − πu, we obtain Moreover, the true error can be divided into a significant part and a less significant part as where and Proof.Adding and subtracting P − h u to u h − πu, we write Taking the L 2 -norm and applying the triangle inequality, we obtain Using the estimates (2.18) and (3.5), we establish (3.20).
Next, we add and subtract πu to e u , we have Furthermore, one can split the interpolation errors u − πu on I i as in (3.3) to obtain Next, we will prove (3.21c).Using the Cauchy-Schwarz inequality and the inequality 2|ab| ≤ a 2 + b 2 , we write for k = 0, 1, 2,

Using the inverse inequality (πu
Summing over all elements and applying (3.4b) and (3.20) yields which gives the estimate (3.21c).
In order to show (3.22), we note that Differentiating (3.24) with respect to x, taking the L 2 -norm, and applying the Cauchy-Schwarz inequality and the inequality |ab| ≤ 1 2 (a 2 + b 2 ), we get Summing over all elements and applying (3.4a) and (3.21c), we obtain

A posteriori error estimation
In this section, we present a technique to compute asymptotically correct a posteriori estimates of the UWDG errors for the nonlinear IVP (1.1).These estimates are computed by solving a local problem with no boundary condition on each element.We further prove that the DG discretization error estimates converge to the true spatial errors in the L 2 -norm as h → 0. Next, we present the weak finite element formulation to compute a posteriori error estimate for the nonlinear IVP (1.1).
Multiplying (1.1) by arbitrary smooth function v and integrating over an arbitrary element I i , we get Replacing u by u h + e u and choosing the test function v = T p+1,i , we obtain Substituting (3.21a), i.e., e u = α i T p+1,i (x) + ω i , into the left-hand side of (4.2) yields Solving for α i and using (2.12b), we obtain Our error estimate procedure consists of approximating the true error on each element I i by the leading term as where the coefficient of the leading term of the error, a i , is obtained from the coefficient α i defined in (4.4) by neglecting the unknown terms ω i and e u , i.e., We note that our error estimates are obtained by solving local problems with no boundary conditions.
An accepted efficiency measure of a posteriori error estimates is the effectivity index.In this paper, we use the global effectivity index and is used to appraise the accuracy of the error estimate.Ideally, the global effectivity index should stay close to one and should converge to one under mesh refinement.
Next, we will show that the error estimate E u converges to the exact error e u in the L 2 -norm as h → 0. Furthermore, we will prove the convergence to unity of the global effectivity index Θ u under mesh refinement.
The main results of this section are stated in the following theorem.In particular, we state and prove asymptotic results of our a posteriori error estimates.
Theorem 4.1.Suppose that the assumptions of Theorem 2.1 are satisfied.If E u = a i T p+1,i (x), x ∈ I i , where a i , i = 1, 2, . . ., N, are given by (4.5b), then there exists a positive constant C independent of h such that Furthermore, we have As a consequence, the UWDG method combined with the a posteriori error estimation procedure yields O(h p+2 ) superconvergent solution i.e., Finally, if there exists a constant C = C(u) > 0 independent of h such that ∥e u ∥ ≥ Ch p+1 , (4.9) then the global effectivity index in the L 2 norm, which is defined as Θ u = ∥Eu∥ ∥eu∥ , converges to unity at O(h) rate i.e., Proof.First, we will prove (4.6).Since e u = α i T p+1,i + ω i and E u = a i T p+1,i on I i , we have where we used the inequality (a + b) 2 ≤ 2a 2 + 2b 2 .Summing over all elements and applying the estimate (3.21c) yields Next, we will estimate N i=1 (α i − a i ) 2 ∥T p+1,i ∥ 2 0,I i .Subtracting (4.5b) from (4.4), we obtain Thus, Using the Lipschitz condition (2.15) and applying the Cauchy-Schwarz inequality yields Squaring both sides and applying the inequality (a + b) 2 ≤ 2(a 2 + b 2 ), we obtain Multiplying by ∥T p+1,i ∥ 2 0,I i and using (2.12a), i.e., ∥T p+1,i ∥ 2 0,I i = h i σp 2 yields Summing over all elements and using h = max Combining this estimate with (2.16) and (3.21c), we get yields which completes the proof of (4.6).
In order to prove (4.7), we use the reverse triangle inequality to have Combining (4.17) and (4.6) completes the proof of (4.7).
Using the relation e u = u − u h and the estimate (4.6), we obtain In order to prove (4.10), we divide the inequality in (4.17) by ∥e u ∥ and we use the estimate (4.6) and the assumption (4.9) to obtain Therefore, ∥Eu∥ ∥eu∥ = 1 + O(h), which completes the proof of (4.10).
In the previous theorem, we proved that the global a posteriori error estimates converge to the true spatial errors at O(h p+2 ) rate.We further proved that the global effectivity index in the L 2 -norm converges to unity at O(h) rate.
Remark 4.1.We note that the computable quantity u h + E u converge to the exact solution u at O(h p+2 ) rate.This accuracy enhancement is simply achieved by adding the error estimate E u to the approximate solution u h .Finally, we note that our DG error estimates are obtained by solving a local problem with no boundary conditions on each element.This leads to very efficient computations of the post-processed approximation u h + E u .
Remark 4.2.The performance of an error estimator is commonly measured by the effectivity index which is the ratio of the estimated error to the actual error.In particular, we say that the error estimator is asymptotically exact if the effectivity index approaches unity as the mesh size goes to zero.Thus, (4.10) indicated that our a posteriori error estimator is asymptotically exact.We note that E u is a computable quantity since it only depends on the numerical solution u h and the f .It provides an asymptotically exact a posteriori estimator on the actual error ∥e u ∥.We would like to emphasize that our DG error estimate is computationally simple which make it useful in adaptive computations.
Remark 4.3.The assumption (4.9) implies that terms of order O(h p+1 ) are present in the error.If this were not the case, the error estimate E u might not be such a good approximation of the error e u .Even though the proof of (4.10) is valid under the assumption (4.9), our computational results given in the next section demonstrate that the global effectivity index in the L 2 -norm converge to unity at O(h) rate.Thus, the proposed error estimation technique is an excellent measure of the error.
We note that the a priori estimate (2.16) is optimal in the sense that the exponent of h is the largest possible.In fact, one may show that provided that the (p + 1)st-order derivatives of the exact solution u do not vanish identically over the domain (u ̸ ∈ V p h ), then an inverse estimate of the form (4.9) is valid [4,13,14] for some positive constant C which depends on u but not on h.Combining (2.16) with (4.9), we show that u h approximates u to O(h p+1 ) in the L 2 norm.

Numerical examples
In this section, we provide several numerical examples to verify our theoretical findings.In our experiments, the system of nonlinear algebraic equations resulting from the UWDG scheme (2.2) is approximated using the Newton-Raphson method.The stopping criterion for Newton's iteration is taken 10 −15 .In all examples, we will compute the L 2 errors ∥u h − πu∥, ∥e u − E u ∥, and |∥e u ∥ − ∥E u ∥|.In all numerical experiments, the numerical order of convergence is computed using the formula − ln(||e N 1 , where e N 1 u and e N 2 u denote the errors using N 1 and N 2 elements, respectively.
For easy visualization, we plot the L 2 -errors in log scale.For each degree p, we fit, in a leastsquares sense, the data sets with a linear polynomial function and then we compute the slope of the fitting line.
Example 5.1.In this example, we consider the following nonlinear problem where the analytical solution is u(x) = e x .We solve (5.1) using the UWDG scheme presented in Section 2. We use the uniform mesh with N = 3,4,. . .,10 elements.We use the finite element spaces V p h with p = 2, 3, 4, 5 to compute the UWDG solution u h ∈ V p h .The L 2 errors ∥u h − πu∥ are reported in the left figure of Figure 1.These results indicate that the UWDG solution u h is O(h p+2 ) superclose to πu.This is in full agreement with the theory.
Next, we implement the proposed error estimation procedure presented in Section 4 to find the error estimator E u on each element and the error ∥e u − E u ∥.In the right figure of Figure 1, we show the L 2 -norm of the errors between e u and E u .These results indicate that ∥e u − E u ∥ is O(h p+2 ) in the L 2 -norm for p = 2, 3, 4, 5.In the left figure of Figure 2, we show the convergence rates for the global errors |∥e u ∥ − ∥E u ∥|.We observe that |∥e u ∥ − ∥E u ∥| = O(h p+2 ) as h → 0. We conclude that our a posteriori error estimate E u converges to the actual error e u as h → 0. These results are in full agreement with the theoretical estimate of Theorem 4.1.Thus, the above experiments show that the orders of convergence given in this paper are sharp.We note that since e u = u − u h , we have e u − E u = u − (u h + E u ).Consequently, we have which shows that the post-processed solution defined by u * h = u h + E u converges to the exact solution u at O(h p+2 ) rate.This accuracy enhancement is achieved by adding the quantity E u to the UWDG solution u h only once at the end of the computation.This leads to a very efficient computation of the superconvergent solution u * h .We present the global effectivity indices Θ u in Table 1.We see that Θ u is near unity and converges to one under h-refinement.Finally, the errors |Θ u − 1| and their orders of convergence shown in the right figure of Figure 2 suggest that |Θ u − 1| is O(h).Thus, the numerical convergence rate is the same as the theoretical rate derived in Theorem 4.1.Example 5.2.In this example, we consider the following problem where the exact solution is u(x) = e x .We solve (5.2) using the UWDG scheme presented in Section 2. We use the uniform mesh with N = 4,5,. . .,12 elements.We compute the UWDG solution u h ∈ V p h with p = 2, 3, 4, 5.We present the L 2 -norm of the errors ∥u h − πu∥ in the left figure of Figure 3.These results indicate that the UWDG solution u h is O(h p+2 ) super close to πu.This is in full agreement with the theory.
Next, we implement the proposed error estimation procedure presented in Section 4 to find the error estimator E u on each element the error u − E u ∥.In the right figure of Figure 3, we show the L 2 -norm of the errors between e u and E u .These results indicate that ∥e u − E u ∥ is O(h p+2 ) in the L 2 -norm for p = 2, 3, 4, 5.In the left figure of Figure 4, we show the convergence rates for the global errors |∥e u ∥ − ∥E u ∥|.We observe that |∥e u ∥ − ∥E u ∥| = O(h p+2 ) as h → 0. We conclude that our a posteriori error estimate E u converges to the actual error e u as h → 0. These results are in full agreement with the theoretical estimate of Theorem 4.1.Thus, the above experiments show that the orders of convergence given in this paper are sharp.
We present the global effectivity indices Θ u in Table 2.We see that Θ u is near unity and converges to one under h-refinement.Finally, the errors |Θ u − 1| and their orders of convergence shown in the right figure of Figure 4 suggest that |Θ u − 1| is O(h).Thus, the numerical convergence rate is the same as the theoretical rate derived in Theorem 4.1.Example 5.3.In this example, we consider the following problem The exact solution is given by u(x) = sin(x).We apply the UWDG scheme presented in Section 2 to (5.3) using uniform mesh with N = 6, 7,. . .,15 elements.We compute the UWDG solution u h ∈ V p h with p = 2, 3, 4, 5.We report the L 2 -norm of the errors ∥u h − πu∥ in the left figure of Figure 5.These results indicate that the UWDG solution u h is O(h p+2 ) superclose to to the interpolant πu.This is in full agreement with the theory.
Next, we implement the proposed error estimation procedure presented in Section 4 to find the error estimator E u on each element and the error ∥e u − E u ∥.In the right figure of Figure 5, we present the L 2 -norm of the errors between e u and E u .These results indicate that ∥e u − E u ∥ is O(h p+2 ) in the L 2 -norm for p = 2, 3, 4, 5.In the left figure of Figure 6, we show the convergence rates for the global errors |∥e u ∥ − ∥E u ∥|.We observe that |∥e u ∥ − ∥E u ∥| = O(h p+2 ) as h → 0. We conclude that our a posteriori error estimate E u converges to the actual error e u as h → 0. These results are in full agreement with the theoretical estimate of Theorem 4.1.Thus, the above experiments show that the orders of convergence given in this paper are sharp.
We present the global effectivity indices Θ u in Table 3.We see that Θ u is near and converges to one under h-refinement.Finally, the errors |Θ u − 1| and their orders of convergence shown in the right figure of Figure 6 suggest that |Θ u − 1| is O(h).Thus, the numerical convergence rate is the same as the theoretical rate derived in Theorem 4.1.

Concluding remarks
In this paper, we proposed and analyzed an a posteriori error estimator for the ultra-weak discontinuous Galerkin (UWDG) method for nonlinear second-order initial-value problems for ordinary differential equations of the form u ′′ = f (x, u).We proved that the significant part of the discretization error for the p-degree UWDG solution is proportional to a (p + 1)-degree Jacobi polynomial.We used these results to construct asymptotically exact a posteriori error estimates.The proposed a posteriori error estimator is computationally simple, efficient, and asymptotically exact.This estimator is obtained by solving a local residual problem on each element.The proposed a posteriori error estimate is shown to converge to the actual error in the L 2 -norm under mesh refinement.The order of convergence is proved to be p + 2, when piecewise polynomials of degree p ≥ 2 are used.We are currently investigating the superconvergence properties and the asymptotic exactness of a posteriori error estimates for UWDG methods applied to two-dimensional elliptic, parabolic, and hyperbolic equations on rectangular and triangular meshes.Our future work will focus on extending our a posteriori error analysis to problems on tetrahedral meshes.